The error in the peak position does depend on the digital resolution of the spectrum (which is the sweep width divided by the number of points). However, the error in the peak position may be less or more than the digital resolution.

For sharp peaks, the error in peak position may be lower than the digital resolution, if the peak position is identified by interpolation (many software packages use interpolation).

For weak and broad peaks, the error in peak position may be larger than the digital resolution because of the difficulty in locating the true location of the signal maximum in the presence of noise.

To understand why this is the case, consider the following set of gedanken experiments for a system with a T2 of 50 msecs with identical sampling rate (10000 pts/s) in the time domain.
a) Acquisition time of 500 msecs and b) Acquisition time of 1000 msecs. The digital resolution of the second experiment (b) would be twice that of the first (a).
However, the error in peak position would be the same or lower in the first experiment. This is because the signal intensity is practically zero at the end of the first 500 msecs (in this case ~10*T2); and the additional acquisition time only adds noise and therefore there is no additional information regarding the signal in the second experiment (b).

Peak position and associated error can be determined accurately by 'curve fitting.' A 'line-shape' is chosen based upon the known relaxation mechanism and an attempt is made to fit the experimental data to the predicted line-shape with position and line-width as variables. In the most favorable case (of infinite digital resolution,correct choice of the line-shape and the absence of noise) the peak-position can be determined with zero error. However, in general, due to the limited digital resolution, especially for multidimensional spectra, and due to the presence of noise, several different values of peak-positions may give rise to equally good fits. The range of these values of peak-positions (that give rise to nearly equally good fit) can be used to determine the error in the peak position. In those cases where the NMR signal has a fine structure (e.g. due to coupling), additional variables such as the coupling constant have to be considered during the fitting process.

~15 years ago I wrote a curve fitting program to extract NMR parameters from 2D spectra, but I do not have that with me now. However,... For 1D, there are many spectral simulation packages that can be used to fit your experimental data. (However, most of these assume idealized line-shapes, so you should make sure that the effects of apodization on line-shape are considered).
For 2D, & 3D spectra you may extract a slice and use the same method. However, the errors will certainly be lower if the information in the entire peak is considered.

Error in peak position is a function of the resolution of your peak. This can be variable based on relaxation, magnetic susceptibility, # of data points, external field homogeneity, and other things. There really is not an "error" in where the NMR peak is as it can only appear where the resonance is (unless of course the peak is folded in or a quad image...). One doesn't measure the peak position like one makes other physical observations so I think reporting the resolution limits based on your data points and sweep width might be sufficient.